Question

Find the N-point DFT of the following sequences x[n] a) x(n) = delta(n) b) x(n) = u(n) - u(n - N) c) x(n) = cos{frac{pi}{2}n} quad ext{for } n = 0, 1, 2, 3 d) x(n) = (frac{1}{2})^n quad ext{for } n = 0, 1, 2, 3 a) and b) 2 marks each c) and d) 3 marks each Hint: 1. For a) and b), use the DFT definition 2. For c) and d), use the geometric sum and cosine formulas given below $sum_{n=0}^{N-1} (x)^n = frac{1 - x^N}{1 - x}$ Geometric sum $cos heta = frac{1}{2} (e^{j heta} + e^{-j heta})$ Cosine formula

          Find the N-point DFT of the following sequences x[n]
a) x(n) = delta(n)
b) x(n) = u(n) - u(n - N)
c) x(n) = cos{frac{pi}{2}n} quad 	ext{for } n = 0, 1, 2, 3
d) x(n) = (frac{1}{2})^n quad 	ext{for } n = 0, 1, 2, 3
a) and b) 2 marks each
c) and d) 3 marks each
Hint:
1. For a) and b), use the DFT definition
2. For c) and d), use the geometric sum and cosine formulas given below
$sum_{n=0}^{N-1} (x)^n = frac{1 - x^N}{1 - x}$ Geometric sum
$cos 	heta = frac{1}{2} (e^{j	heta} + e^{-j	heta})$ Cosine formula

$Find the N-point DFT of the following sequences x[n] a) x(n) = delta(n) b) x(n) = u(n) - u(n - N) c) x(n) = cosfracpi2n quad extfor n = 0, 1, 2, 3 d) x(n) = (frac12)^n quad extfor n = 0, 1, 2, 3 a) and b) 2 marks each c) and d) 3 marks each Hint: 1. For a) and b), use the DFT definition 2. For c) and d), use the geometric sum and cosine formulas given below sumn=0^N-1 (x)^n = frac1 - x^N1 - x Geometric sum cos heta = frac12 (e^j heta + e^-j heta) Cosine formula$

Added by Aitor H.

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